Download 8.4 A Significance Test for the Difference of Two Proportions

Section 8.4
Significance Test for the
Difference of Two
Proportions
Critical Values
For a 2-sided test, what are the critical
values for a significance level of α = 0.05?
Critical Values
For a 2-sided test, what are the critical
values for a significance level of α = 0.05?
Critical Values
For a 2-sided test, what are the critical
values for a significance level of α = 0.05?
Critical Values
For a 2-sided test, what are the critical
values for a significance level of α = 0.05?
invNorm (.025) = -1.96
invNorm (.975) = 1.96
z* = ± 1.96
Critical Values
For a one-sided test, what is the critical
value for a significance level of α = 0.05?
Critical Values
For a one-sided test, what is the critical
value for a significance level of α = 0.05?
Critical Values
For a one-sided test, what is the critical
value for a significance level of α = 0.05?
If Ha: p < po, invNorm (.05 ) = - 1.64
Critical Values
For a one-sided test, what is the critical
value for a significance level of α = 0.05?
Critical Values
For a one-sided test, what is the critical
value for a significance level of α = 0.05?
If Ha: p > po, invNorm (.95 ) = 1.64
Components of a Significance Test for the
Difference of Two Proportions for Surveys
Four parts:
Components of a Significance Test for the
Difference of Two Proportions for Surveys
Four parts:
1. Name test and check conditions
Components of a Significance Test for the
Difference of Two Proportions for Surveys
Four parts:
1. Name test and check conditions
2. Write null and alternative hypotheses
Components of a Significance Test for the
Difference of Two Proportions for Surveys
Four parts:
1. Name test and check conditions
2. Write null and alternative hypotheses
3. Compute test statistic and P-value
Components of a Significance Test for the
Difference of Two Proportions for Surveys
Four parts:
1. Name test and check conditions
2. Write null and alternative hypotheses
3. Compute test statistic and P-value
4. Write a conclusion in context
Name Test & Check Conditions
Name: Use the whole name!
One-sided significance test for the difference
of two proportions
or
Two-sided significance test for the difference
of two proportions
Check Conditions
First condition:
Random samples selected independently
from two different populations
Check Conditions
Second condition: normal?
All these quantities must be at least 5.
Show all 4 calculations and state
significance of results
Check Conditions
Third condition:
Each population is at least 10 times as large
as its sample size.
Remember to explain why this is
reasonable.
Null Hypothesis
Null hypothesis almost always one of “no
difference” or “no effect”
Null Hypothesis
Null hypothesis almost always one of “no
difference” or “no effect”
Several choices as long as define p1 and p2
Ho: The proportion of successes p1 in the
first population is equal to the proportion
of successes p2 in the second population.
Write Ho in context of situation.
Null Hypothesis
Null hypothesis almost always one of “no
difference” or “no effect”
Ho: p1 = p2, where p1 is the proportion of
successes in the first population and p2 is
the proportion of successes in the second
population.
Write Ho in context of situation.
Null Hypothesis
Null hypothesis almost always one of “no
difference” or “no effect”
Ho: p1 - p2 = 0, where p1 is the proportion of
successes in the first population and p2 is
the proportion of successes in the second
population.
Write Ho in context of situation.
Alternative Hypothesis
Form depends on whether you
need a two-sided or one-sided
test.
Alternative Hypothesis
Two-sided test
Ha: The proportion of successes p1 in the
first population is not equal to the
proportion of successes p2 in the second
population.
In symbols:
Ha: p1 ≠ p2 or Ha: p1 – p2 ≠ 0
Alternative Hypothesis
One-sided test
Ha: The proportion of successes p1 in the
first population is greater than the
proportion of successes p2 in the second
population.
In symbols:
Ha: p1 > p2 or Ha: p1 – p2 > 0
Alternative Hypothesis
One-sided test
Ha: The proportion of successes p1 in the
first population is less than the proportion
of successes p2 in the second population.
In symbols:
Ha: p1 < p2 or Ha: p1 – p2 < 0
Compute Test Statistic and P-value
If we assume p1 = p2, then use
Compute Test Statistic and P-value
If we assume p1 = p2, then use
Compute Test Statistic and P-value
If we assume p1 = p2, then use
is called “pooled estimate” of the
common proportion of successes
total
number
of
successes
in
both
samples

Compute Test Statistic and P-value
or use 2-PropZTest
Only valid if we assume p1 = p2
Compute Test Statistic and P-value
If do not assume p1 = p2, then use:
Compute Test Statistic and P-value
The P-value is the probability of getting a
value of z as extreme or more extreme
than that from your samples if Ho is true.
Include a sketch.
Write a Conclusion in Context
State whether you reject or do not reject
the null hypothesis.
Write a Conclusion in Context
State whether you reject or do not reject the
null hypothesis.
Link this conclusion to your computations by
(1) comparing z to critical value z*
– reject null hypothesis if z is more extreme
than z*
or
Write a Conclusion in Context
State whether you reject or do not reject the null
hypothesis.
Link this conclusion to your computations by
(1) comparing z to critical value z*
– reject null hypothesis if z is more extreme than z* or
(2) comparing P-value to level of
significance α
– reject null hypothesis if P-value is smaller
than α
Write a Conclusion in Context
State whether you reject or do not reject the null
hypothesis.
Link this conclusion to your computations by (1)
comparing z to critical value z*, reject null
hypothesis if z is more extreme than z* or
(2) comparing P-value to level of significance α,
reject null hypothesis if P-value is smaller
than α
Write sentence giving conclusion in
context related to alternative
hypothesis.
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Page 536, P50
Give appropriate statistical evidence to
support your answer means complete all 4
steps for a significance test.
Page 536, P50
Name: one-sided significance test for the
difference of two proportions
because you are asked whether the data
support the conclusion that there was a
decrease in voter support for the
candidate.
Page 536, P50
We are told that we have two random
samples from a population of probable
voters in some city.
It’s reasonable to assume that the samples
are independent because the random
samples were taken about 2 weeks apart.
Page 536, P50
Let n1 be 1st sample and n2 the 2nd sample.
Each of the following is at least 5:
Page 536, P50
The number of probable voters at both times
should be larger than 10 times the sample
size for both samples, and it is unknown
whether this is the case.
Page 536, P50
Using symbols, what is the null hypothesis?
Page 536, P50
Ho: p1 = p2
Page 536, P50
Ho: p1 = p2, where p1 is the proportion of all
probable voters who favored the candidate
at the time of the first survey and p2 is the
proportion of all probable voters who
favored the candidate one week before the
election.
Page 536, P50
Ha: p1 > p2
We want to see if there was a decrease in
voter support from 1st survey to 2nd survey.
Test Statistic and P-value
2-PropZTest
x1: 321
n1: 600
x2: 382
n2: 750
p1 > p2
Calculate
Survey 3 weeks before
start of campaign
2nd survey
Test Statistic and P-value
2-PropZTest
x1: 321
Survey 3 weeks before
start of campaign
n1: 600
x2: 382
2 survey
n2: 750
p1 > p2
Calculate
z = 0.938 and P-value = 0.1741
nd
Page 536, P50
Write Conclusion in Context
I do not reject the null hypothesis because
the P-value of 0.1741 is greater than the
significance level of α = 0.05.
Write Conclusion in Context
I do not reject the null hypothesis because
the P-value of 0.1741 is greater than the
significance level of α = 0.05.
There is not sufficient evidence to
conclude that there was a decrease in
voter support for the new candidate after
the parking tickets were revealed.
For this one-sided test:
What critical value(s) of z would you
use to compare against the test
statistic of z = 0.938 for a 95%
confidence level?
Z*
What critical value(s) of z would you use to
compare against the test statistic of z =
0.938 for a 95% confidence level?
One-sided test at 95% confidence level:
z* = invNorm(0.95) = 1.64
Write Conclusion in Context
I do not reject the null hypothesis because
the test statistic of z = 0.938 is less
extreme than the critical value of z* = 1.64.
There is not sufficient evidence to
conclude that there was a decrease in
voter support for the new candidate after
the parking tickets were revealed.
Problem # 1
If the P-value of a test is less than the
level of significance, then which of
these conclusions is correct?
If the P-value of a test is less than the level of significance,
then which of these conclusions is correct?
A. The value of the test statistic is in the
rejection region for this test.
B. The sample size should be increased to
decrease the margin of error.
C. The null hypothesis is true.
D. The corresponding confidence interval
will contain the hypothesized value of the
parameter in the null hypothesis.
E. None of these is a valid conclusion.
If the P-value of a test is less than the level of significance,
then which of these conclusions is correct?
A. The value of the test statistic is in
the rejection region for this test.
B. The sample size should be increased to
decrease the margin of error.
C. The null hypothesis is true.
D. The corresponding confidence interval
will contain the hypothesized value of the
parameter in the null hypothesis.
E. None of these is a valid conclusion.
Problem # 1
A. If the P-value is less than α, then the
result is “statistically significant.” Reject
H0.
This corresponds to the test statistic falling
in the rejection region.
If all else remains the same, which of these
will make a confidence interval for the
difference of two proportions wider?
I. Increase the confidence level
II. Increase the sample size
III. Increase the margin of error
IV. Increase the probability of a Type I error
If all else remains the same, which of these
will make a confidence interval for the
difference of two proportions wider?
I. Increase the confidence level
II. Increase the sample size
III. Increase the margin of error
IV. Increase the probability of a Type I error
One-sided or Two-sided?
Page 537
E64
E66
E67
E68
One-sided or Two-sided?
E64: Two-sided
E66:
E67:
E68:
One-sided or Two-sided?
E64: Two-sided
E66: Two-sided
E67:
E68:
One-sided or Two-sided?
E64: Two-sided
E66: Two-sided
E67: One-sided
E68:
One-sided or Two-sided?
E64: Two-sided
E66: Two-sided
E67: One-sided
E68: One-sided
Questions?